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http://functions.wolfram.com/10.08.06.0004.02
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PolyLog[\[Nu], z] \[Proportional] Zeta[\[Nu]] + Zeta[-1 + \[Nu]] (z - 1) +
(1/2) (Zeta[-2 + \[Nu]] - Zeta[-1 + \[Nu]]) (z - 1)^2 + \[Ellipsis] /;
(z -> 1) && Re[\[Nu]] > 1 && !Element[\[Nu], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Zeta", "[", "\[Nu]", "]"]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], " ", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", TagBox["\[Nu]", Rule[Editable, True]], ")"]]]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> + </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", TagBox[RowBox[List["\[Nu]", "-", "1"]], Rule[Editable, True]], ")"]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["\[Nu]", "-", "2"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["\[Nu]", "-", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∉ </mo> <mi> ℤ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", TagBox["\[Nu]", Rule[Editable, True]], ")"]]]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> + </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", RowBox[List["(", TagBox[RowBox[List["\[Nu]", "-", "1"]], Rule[Editable, True]], ")"]]]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["\[Nu]", "-", "2"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["\[Nu]", "-", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∉ </mo> <mi> ℤ </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "\[Nu]", "]"]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "1"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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